A solid cylindrical block of radius 12 cm and height 18 cm is mounted with a conical block of radius 12 cm and height 5 cm. The total lateral surface of the solid thus formed is

### Concept & Lateral Surface Area The total lateral surface area (LSA) of a composite solid is the sum of the lateral surface areas of its exposed individual parts. $$ \text{LSA}_{\text{cylinder}} = 2 \pi r h $$ $$ \text{LSA}_{\text{cone}} = \pi r l $$ where $l = \sqrt{r^2 + h^2}$. ### Step-by-Step Solution - For the cylinder: radius $r = 12$ cm, height $h_1 = 18$ cm. - Lateral surface area of the cylinder = $2 \pi r h_1 = 2 \times \frac{22}{7} \times 12 \times 18$. - For the cone: radius $r = 12$ cm, height $h_2 = 5$ cm. - First, find the slant height $l$ of the cone: $l = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ cm. - Lateral surface area of the cone = $\pi r l = \frac{22}{7} \times 12 \times 13$. - Total lateral surface area = $2 \pi r h_1 + \pi r l = \pi r (2h_1 + l)$. - Substituting the values: $\frac{22}{7} \times 12 \times (2(18) + 13)$. - Calculate inside the parentheses: $36 + 13 = 49$. - Total area = $\frac{22}{7} \times 12 \times 49 = 22 \times 12 \times 7 = 1848$ cm$^2$. ### Exam Strategy & Shortcut Factor out common terms to avoid multiplying by $\frac{22}{7}$ twice. Factoring out $\pi r$ gives $\pi r(2h_1 + l)$, which simplifies the multiplication to $\frac{22}{7} \times 12 \times 49$. The $49$ easily cancels with the $7$ in the denominator. ### Common Pitfall Adding the area of the base(s). The question specifically asks for the "lateral surface", meaning the curved surface area only, excluding the top and bottom circular bases. ### Final Answer Therefore, the correct answer is **1848 cm$^2$**.